Empirical likelihood test for high-dimensional two-sample model

Abstract

A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null hypothesis is proved, a result which is different from the χ 2 law for a model with a fixed variable number. Under alternative hypothesis, the test statistic diverges. We can then find the asymptotic confidence region for the difference of parameters of the two phases. Some Monte-Carlo simulations study the behaviour of the proposed test statistic.Keywords Two-sample · high-dimension · linear model · empirical likelihood test. IntroductionThe technology development and fast numerical techniques make possible to consider and study statistical models with a large number of variables. High-dimensional model refers to a model whose the number p of explanatory variables increases to infinity as the number n of observations converges to infinity. When p diverges, traditional statistical methods may not work with this kind of growth dimensionality.Most of the literature works on high-dimensional model utilize the LASSO (Least Absolute Shrinkage and Selection Operator) type methods, in order to automatically select the significant variables. The principle of these methods, introduced by Tibshirani (1996), is to optimize a penalized process, more precisely, a process with a L 1 -type penalty. If the model contains outliers, the parameter estimators by the least squares method with LASSO penalty have a large error. An alternative method is then the penalized quantile method. Thereby, Dicker et al. (2013) consider a quantile model with seamless-L 0 penalty when the number p of explanatory variables is such that p → ∞, p/n → 0 as n → ∞. For a general quantile regression, Wu and Liu (2009) propose the SCAD penalty, while, in Zou andYuan (2008), a composite quantile regression is considered with an adaptive LASSO penalty. The case p → ∞ is also considered in Fan and Peng (2004) for a non-concave penalized likelihood method, when p 5 /n → ∞. Concerning the group selection methods for high-dimensional models, the readers find in Huang et al. (2012) a review of methods. All these methods are based first on the principle of selecting (automatically) the significant variables. Then, the dependent variable is modeled only as a function of the significant variables, in order to have more accurate parameter estimators and a better adjustment for the dependent variable.

A review of recent advances in empirical likelihood

“…Zang et al (2016) introduced the jackknife empirical likelihood test for high-dimensional regression coefficients. Ciuperca and Salloum (2016) proposed an empirical likelihood test for the high-dimensional two-sample model. Wang et al (2013) developed a jackknife empirical likelihood test for the equality of two high-dimensional means.…”

Personal Privacy Protection via Irrelevant Faces Tracking and Pixelation in Video Live Streaming

“…Namely, we cannot impose f = f G because: (i) the amount of proposed faces within a segment is not significantly larger than the dimension number of the face vectors; (ii) the amount of proposed faces within a segment is insufficient in guaranteeing the proposed face sample complies with a Gaussian distribution. Inspired by [50], for better robustness and avoiding parametric-imposing, we propose a two-sample test to reject the proposed false positives based on empirical likelihood ratio (ELR) statistics.…”

“…x i and y j are the elements in z and z after the projection. Referring [50], [52], H 0 can be written as:…”